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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 07}
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%\date{2021 年 2 月 28 日}
\date{May 18, 2021}
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%\subsection{Homework 07}
%E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

\begin{document}

\maketitle

\begin{enumerate}

\item [E7.1.2.] Consider a renewal process in which the inter-occurrence times have an exponential distribution with parameter $\lambda$:
%\begin{eqnarray*}
$f(x)=\lambda e^{-\lambda x}, \text{ and } F(x) = 1- e^{-\lambda x} \text{ for } x > 0.
$ %\end{eqnarray*}
Calculate $F_2(t)$ by carrying out the appropriate convolution and then determine $\mathbb{P}\{N(t) = 1\}$.


\item [E7.1.3.] Which of the following are true statements?
\begin{enumerate}
\item  $N(t) < k$ if and only if $W_k>t$. 
\item  $N(t) \le k$ if and only if $W_k \ge t$. 
\item  $N(t) > k$ if and only if $W_k < t$.
\end{enumerate}


\item [E7.2.1.] Let $\{X_n; n=0,1,\cdots\}$ be a two-state Markov chain with the transition probability matrix
\begin{eqnarray*}
P=
\begin{blockarray}{ccc}
& 0 & 1  \\
\begin{block}{c[cc]}
  0 & 1-a  & a   \\
  1 & b     & 1-b \\ 
\end{block}
\end{blockarray}.
\end{eqnarray*}
State 0 represents an operating state of some system, while state 1 represents a repair state. 
We assume that the process begins in state $X_0 = 0$, and then the successive returns to state 0 from the repair state form a renewal process. 
Determine the mean duration of one of these renewal intervals.


\item [E7.2.3.] Calculate the mean number of renewals $M(n)=\mathbb{E}[N(n)]$ for the renewal process having inter-occurrence distribution $p_1 =0.4$, $p_2 =0.1$, $p_3 =0.3$, $p_4 =0.2$ for $n = 1,2,...,10$. Also calculate $u_n = M(n) - M(n-1)$.


\item [P7.2.1.] For the block replacement example of this section for which $p_1 = 0.1$, $p_2 = 0.4$, $p_3 = 0.3$, and $p_4 = 0.2$, suppose the costs are $c_1 = 4$ and $c_2 = 5$. Determine the minimal cost block period $K^*$ and the cost of replacing upon failure alone.



\end{enumerate}


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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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